C. Theorems of Continuity
THEOREM–1 If f & g are two functions that are continuous at x= c then the functions defined by F_{1}(x) = f(x) ± g(x) ; F_{2}(x) = K f(x) K any real number ; F_{3}(x) = f(x).g(x) are also continuous at x= c.
Further, if g (c) is not zero, then is also continuous at x = c.
THEOREM–2 If f(x) is continuous & g(x) is discontinuous at x = a then the product function ø(x) = f(x) . g(x) is not necessarily discontinuous at x = a .
THEOREM–3 If f(x) and g(x) both are discontinuous at x = a then the product function ø(x) = f(x) . g(x) is not necessarily discontinuous at x = a .
Theorems4 : Intermediate Value Theorem
If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k.
Note that the Intermediate Value Theorem tells that at least one c exists, but it does not give a method for finding c. Such theorems are called existence theorems.
As a simple example of this theorem, consider a person's height. Suppose that a girl is 5 feet tall on her thirteenth birthday and 5 feet 7 inches tall on her fourteenth birthday. Then, for any height h between 5 feet and 7 inches, there must have been a time t when her height was exactly h. This seems reasonable because human growth is continuous and a person's height does not abruptly change from one value to another.
The Intermediate Value Theorem guarantees the existence of at least one number c in the closed interval [a, b]. There may, of course, be more than one number c such that f(c) = k, as shown in Figure 1. A function that is not continuous does not necessarily possess the intermediate value property. For example, the graph of the function shown in Figure 2 jumps over the horizontal line given by y = k and for this function there is no value of c in [a, b] such that f(c) = k.
The Intermediate Value Theorem often can be used to locate the zeroes of a function that is continuous on a closed interval. Specifically, if f is continuous on [a, b] and f(a) and f(b) differ in sign, then the intermediate Value Theorem guarantees the existence of at least one zero of f in the closed interval [a, b].
Ex.10 Use the Intermediate Value Theorem to show that the polynomial function f(x) = x^{3} + 2x  1 has a zero in the interval [0, 1]
Sol. Note that f is continuous on the closed interval [0, 1]. Because
f(0) = 0^{3} + 2(0)  1 = 1 and f(1) = 1^{3} + 2(1)  1 = 2
it follows that f(0) < 0 and f(1) > 0. You can therefore apply the Intermediate Value Theorem to conclude that there must be some c in [0, 1] such that f(c) = 0, as shown in Figure 3.
Ex.11 State intermediate value theorem and use it to prove that the equation has at least one real root.
Sol. Let f (x) = first, f (x) is continuous on [5, 6]
Hence by intermediate value theorem É at least one value of c ∈ (5, 6) for which f (c) = 0
c is root of the equation
Ex.12 If f(x) be a continuous function in then prove that there exists point
Sol.
Let g(x) = f(x) – f(x + π) ....(i)
at x = π; g(π) = f(π) – f(2π) ....(ii)
at x = 0, g(0) = f(0) – f(π) ...(iii)
adding (ii) and (iii), g(0) + g(π) = f(0) – f(2π)
⇒ g(0) + g(π) = 0 [Given f(0) = f(2π) ⇒ g(0) = –g(π)
⇒ g(0) and g(π) are opposite in sign.
⇒ There exists a point c between 0 and p such g(c) = 0 as shown in graph;
From (i) putting x = c g(c) = f(c) – f(c +π) = 0 Hence, f(c) = f(c + π)
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1. What is the definition of continuity in mathematics? 
2. How do you prove continuity using the epsilondelta definition? 
3. What is the Intermediate Value Theorem? 
4. How does the BolzanoWeierstrass Theorem relate to continuity? 
5. Can a function be continuous at some points and discontinuous at others? 

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